The tan is negative in the 2nd and 4th quadrants, knowing this, the angle that equals the value that tan = 1 in the first quadrant is the tan 45° = 1, but the question restrict for 3pi/2 < theta < 2pi, so it's the same as 270° < theta < 360° (4° Quadrant), then moving the tan 45° = to this quadrant the value will be negative, tan in 4°Q = tan(360 - 45=315°) = -1.
sec theta = 1/cos theta
cos 45° =
[tex] \sqrt{2} \div 2[/tex]
cos values in 4°Q will be positives, so it doesn't change the signal.
sec 315° = 1/cos 315°
sec 315° = 1/square root (2)/2
sec 315 = 1/1 x 2/sqroot(2)
[tex]sec \: 315 = 2 \div \sqrt{2} [/tex]
[tex]sec \: 315 = (2 \div \sqrt{2}) \: \times ( \sqrt{2} \div \sqrt{2})[/tex]
[tex]sec \: 315 = 2 \sqrt{2} \div 2[/tex]
[tex]sec \: 315 = \sqrt{2} [/tex]
